For a two-dimensional incompressible flow field given by $- \to U = A (x^i - y^j)$ , where A>0, which one of the following statements is FALSE?
A. It satisfies continuity equation
B. It is unidirectional when x $\to$ 0 and y $\to \infty$ .
C. Its streamlines are given by x = y.
D. It is irrotational

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Solution

The correct option is C C
C is the false statement
2D incompressible flow continuity equation.
$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$
$\frac{\partial (A x)}{\partial x} + \frac{\partial (- A y)}{\partial y} = 0$
$A - A = 0$ it satisfies continuity equation.
$\Rightarrow A s - \to U = A x^i - A y^j$
where, u=Ax and v= -Ay
$\Rightarrow$ stream line equation for 2D

$\frac{d x}{u} = \frac{d y}{v}$

$\frac{d x}{A x} = \frac{d y}{- A y}$

Inx = -In y+ In c
Inxy = In c
xy = c $\to$ streamline equation
x $\to$ 0, Flow will be along y-axis
y $\to$ 0, Flow will be along x-axis

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